9 problems

Let a Jordan curve be a simple closed curve $\gamma\subset\mathbb{R}^2$. An isosceles trapezoid is a quadrilateral whose vertices are the intersection set of two parallel lines and…

Applicability conjecture. The approach developed in the paper is applicable to Jordan curves satisfying this condition.

Let $f\colon S^2\to S^2$ be a Thurston map with a hyperbolic orbifold that is realized by a rational map. Let $P_f$ denote its postcritical set, and let pullbacks be considered up…

Let $(\mathcal X,\mathcal B,\angle)$ be an angle space conformal to a Jordan curve, meaning that it arises from the angle structure induced by a Jordan curve under conformal equiva…

Let $\gamma$ be a Jordan curve in the plane. An inscribed rectangle is a rectangle whose four vertices lie on $\gamma$, and its aspect ratio is the ratio of its side lengths, taken…

Converse extremality conjecture. The converses of these implications are true: every maximal element of $\mathcal{J}$ contains no open points of $\mathcal{D}$, and every minimal el…

Let $\mathcal{J}^{1}_{\textnormal{pw}}$ be the class of piecewise $C^1$ Jordan curves without cusps, and let $\mathcal{J}^{0}$ be the class of continuous Jordan curves. Let…

A Jordan curve is a simple closed curve. A rectangle has a prescribed aspect ratio when the ratio of its side lengths is fixed. Rectangular Peg conjecture. Every Jordan curve conta…

Average-distance conjecture. For any $k$-touching family $\mathcal{F}$ of Jordan curves, the average distance in $\mathcal{F}$ is at most